2 Answers
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Here is a somewhat standard presentation that might be taught in intermediate macroeconomics at US universities.

Let $p_t$ be the price of a unit of capital at the end of period $t$. Suppose that production takes place at the start of each period. After production takes place, depreciation occurs. Let us say that the depreciation rate is constant at $\delta$. Let us also say that the real interest rate across any two consecutive periods is $r$. Assume also that the price of each unit of output is equal to $1.

If so, we would expect that

$p_1=\frac{1}{1+r}[MP^k_2+(1-\delta)p_2]$.

This has a nice interpretation. The price of capital at the end of period 1 equals: The present discounted value of the additional output it can give us at the start of period 2, plus the after-depreciation resale value of that unit of capital at the end of period 2.

We can write down an analogous expression for $p_2$:

$p_2=\frac{1}{1+r}[MP^k_3+(1-\delta)p_3]$

Now substitute this latter expression for $p_2$ into our first equation above, to get:

This last expression has again a nice interpretation. The price of capital at the end of period 1 equals: The present discounted value of the additional output it can give us at the start of period 2, plus the additional output it can give us at the start of period 3, plus the resale value of that unit of capital at the end of period 3.

If we keep going, plugging in expressions for $p_3$, $p_4$, ..., eventually we'll find that

This has the desired result: The price of a unit of capital today is simply equal to the present value of the future stream of income that this unit of capital will generate, appropriately taking into account depreciation.

This generalizes to a broader principle employed not just in standard macroeconomics but in the real world too: The price of any asset should be equal simply to the present value of the future stream of income that the asset will generate.

Let's think about a unit of non-depreciating capital that has a price of $p_k$, in a static environment where we normalized the price of the consumption good to 1. Furthermore, the unit of capital gives you $r$ units of interest in consumption good units.

tl;dr: Yes, Because of some equilibrium condition.

The value of capital

Let me denote by $\lambda_t$ the value of a marginal consumption good in unit $t$ - this is how much more happy you would be today, if you knew that in time $t$ you get to consume a little bit more.

Then, the utility that a unit of capital generates for you at time $0$ is

$$\lambda_1 r + \lambda_2 r + \lambda_3 r + \dots$$

We assume that capital doesn't give interest in the first period. Note that discount rates are hidden inside $\lambda$; this sequence converges by assumption. This is your discounted real expected return on capital goods.

Cost of Capital

Now, what is the opportunity cost of investing into one unit of capital? You forgo $p_k$ units of consumption in period 0.

Equilibrium Condition

The cost has to equal the value in equilibrium - the price has to equal the value of capital, otherwise:

if the price was higher than the value, people who owned capital would try to sell it

if the price was lower than the value, demand would be larger than supply of capital goods